If \(F̏ := (X, F)\)  is a Bishop space, then \(F̏^* := (X, F^*)\)  is the Bishop space of
bounded elements of \(F\) . The constant functions
\(\mathrm {Const}(X)\)  is the trivial
topology on \(X\) , while \(F̥(X)\)  is the discrete topology on \(X\) . Clearly, if \(F\)  is a topology on \(X\) ,
then \(\mathrm {Const}(X) \subseteq F \subseteq F̥(X)\) , and the set of its bounded elements
\(F^{*}\)  is also a topology on \(X\) . It is straightforward to see that the pair \(Ȑ := ({\mathbb {R}}, {\mathrm {Bic}({\mathbb {R}})})\)  is a
Bishop space, which we call the Bishop space of reals. If \(X\)  is a metric space, the set
\(C_{p}(X)\)  of all weakly continuous functions of
type \(X \rightarrow {\mathbb {R}}\) , as it is defined in [1]}, p.76, is the set of pointwise continuous ones.
It is easy to see that the pair \(\mathcal {W}(X) = (X, C_{p}(X))\) 
is Bishop space. Bishop calls \(C_{p}(X)\)  the weak topology on \(X\) , but here we avoid this term, since in [2]}
we use this term for the Bishop topology that corresponds to the weak topology of open sets,
and we call \(C_{p}(X)\)  the pointwise topology on \(X\) . If \(X\)  is a compact metric space,
the set \(C_{u}(X)\)  of all uniformly continuous
functions of type \(X \rightarrow \mathbb {R}\)  is a topology, called by Bishop the
uniform topology on \(X\) . We call \(\mathcal {U}(X) = (X, C_{u}(X))\)  the uniform space.
If \(X\)  is a locally compact metric space, the set \(\mathrm {Bic}(X)\)  of
Bishop continuous functions from \(X\)  to \({\mathbb {R}}\)  i.e., uniformly continuous on
everyAs in the case of \({\mathrm {Bic}({\mathbb {R}})}\) , it seems that this definition requires quantification over the
power set of \(X\)  i.e.,
\({\mathrm {Bic}(X)}(f) \Leftrightarrow \forall _{B \in \mathcal {P}(X)}(\mbox{bounded}(B)\Rightarrow f_{|B} \ \mbox{is uniformly continuous}).\) 
A bounded subset \(B\)  of an inhabited metric space \(X\)  is a triplet
\((B, x_{0}, M)\) , where \(x_{0} \in X, B \subseteq X\) , and \(M > 0\)  is a bound for \(B \cup \lbrace x_{0}\rbrace \) .
To avoid such a quantification, if \(x_{0}\)  inhabits \(X\) , then for every bounded subset
\((B, x_{0}{^{\prime }}, M)\)  of \(X\)  we have that there is some \(n \in {\mathbb {N}}\)  such that \(n > 0\)  and
\(B \subseteq [d_{x_{0}} \le \overline{n}] = \lbrace x \in X \mid d(x_{0}, x) \le n\rbrace \) .
If \(x \in B\) , then \(d(x, x_{0}) \le d(x, x_{0}{^{\prime }}) + d(x_{0}{^{\prime }}, x_{0}) \le M + d(x_{0}{^{\prime }}, x_{0})\) , therefore
\(x \in [d_{x_{0}} \le \overline{n}]\) , for some \(n > M + d(x_{0}{^{\prime }}, x_{0})\) . Hence,
\({\mathrm {Bic}(X)}(f) \Leftrightarrow \forall _{n \in {\mathbb {N}}}(f_{|[d_{x_{0}} \le \overline{n}]} \ \mbox{is uniformly continuous}),\) 
since \([d_{x_{0}} \le \overline{n}] = \lbrace x \in d(x_0, x) \le n\rbrace \)  is trivially a bounded subset of \(X\) .

bounded subset of \(X\) , is a Bishop topology on \(X\) .
